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arXiv · 2210.06530

An Improvement of the lower bound for the minimum number of link colorings by quandles

Hamid Abchir·Soukaina Lamsifer
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Abstract

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component.

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Hamid Abchir, Soukaina Lamsifer. 2022-10-12. An Improvement of the lower bound for the minimum number of link colorings by quandles. https://arxiv.org/abs/2210.06530

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